J. Chem. Sci., Vol. 121, No. 5, September 2009, pp. 823–837. © Indian Academy of Sciences.
Computing magnetic anisotropy constants of single molecule
magnets†
S RAMASESHA1,*, SHAON SAHOO2, RAJAMANI RAGHUNATHAN1 and
DIPTIMAN SEN3
1
Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560 012
Department of Physics, Indian Institute of Science, Bangalore 560 012
3
Center for High Energy Physics, Indian Institute of Science, Bangalore 560 012
e-mail: ramasesh@sscu.iisc.ernet.in
2
Abstract. We present here a theoretical approach to compute the molecular magnetic anisotropy
parameters, DM and EM for single molecule magnets in any given spin eigenstate of exchange spin Hamiltonian. We first describe a hybrid constant MS-valence bond (VB) technique of solving spin Hamiltonians employing full spatial and spin symmetry adaptation and we illustrate this technique by solving the
exchange Hamiltonian of the Cu6Fe8 system. Treating the anisotropy Hamiltonian as perturbation, we
compute the DM and EM values for various eigenstates of the exchange Hamiltonian. Since, the dipolar
contribution to the magnetic anisotropy is negligibly small, we calculate the molecular anisotropy from
the single-ion anisotropies of the metal centers. We have studied the variation of DM and EM by rotating
the single-ion anisotropies in the case of Mn12Ac and Fe8 SMMs in ground and few low-lying excited
states of the exchange Hamiltonian. In both the systems, we find that the molecular anisotropy changes
drastically when the single-ion anisotropies are rotated. While in Mn12Ac SMM DM values depend
strongly on the spin of the eigenstate, it is almost independent of the spin of the eigenstate in Fe8 SMM.
We also find that the DM value is almost insensitive to the orientation of the anisotropy of the core
Mn(IV) ions. The dependence of DM on the energy gap between the ground and the excited states in both
the systems has also been studied by using different sets of exchange constants.
Keywords. Single molecule magnets; single-ion anisotropy; anisotropy parameters.
1.
Introduction
There has been a widespread interest to develop
molecule-based systems with magnetic ground state
called single molecule magnets (SMMs). Following
the synthesis of Mn12Ac and Fe 8 clusters with highspin ground state, exotic properties like Quantum
Resonant Tunneling (QRT) and quantum coherence
were unveiled.1–4 SMMs are characterized by a high
spin ground state and large negative magnetic anisotropy (DM) which creates an energy barrier between
the states corresponding to positive and negative MS
values (figure 1).5 In this case, the magnetic anisotropy is said to be uniaxial and the ground state of
the molecule would correspond to the highest possible magnetization with total spin SGS. The second
order transverse or rhombic anisotropy mixes various states with total spin SGS that differ in their MS
value by two. Thus, in SGS = 10 ground state, the
†
Dedicated to the memory of the late Professor S K Rangarajan
*For correspondence
MS = +10 and MS = −10 states are connected through
the EM term in the tenth order. EM will be non-zero
only if the SX2 − SY2 term remains invariant under the
symmetry of the molecule. However, fourth order
anisotropy terms need to be included in the anisotropy Hamiltonian for Mn12Ac in which the molecular symmetry prohibits a non-zero EM but still QRT
Figure 1. Schematic of the double potential well for the
SGS = 10 ground state when DM is negative.
823
S Ramasesha et al
824
is observed. The requirement of large negative value
of DM and a large non-zero ground state spin are
stringent requirements for a molecule to behave as a
SMM. Modelling magnetic anisotropy in these systems becomes necessary for developing new SMMs
with desired properties.
Magnetic anisotropy of SMMs is computed by
treating the anisotropy Hamiltonian as a perturbation
over the Heisenberg exchange Hamiltonian, since
the magnitude of anisotropy constants are very small
compared to the exchange constants. SMMs contain
many spin centers with equal or unequal magnitude
of site spins and in most cases the magnetic exchange is frustrated. The Fock space of the Hamiltonian for even the well known SMMs are generally
very large (hundred million in case of Mn12 Ac6 and
more than two billion in the case of Fe12 wheel7) and
obtaining even a few low-lying states of the Hamiltonian could pose a serious computational challenge.
Since the exchange Hamiltonian conserves both
total spin and z-component of total spin (MS), the
problem can be simplified by specializing the basis.
Hamiltonian could be block diagonalized in spinadapted or constant MS basis, so that one can work
with a basis set with a particular total spin or MS
value. This can be further simplified by exploiting
spatial symmetries of the model. An ideal situation
would correspond to one in which all the spin and
spatial symmetries are utilized to construct a fully
symmetrized basis to minimize the size of the Hamiltonian matrix that needs to be diagonalized.
Besides the computational efficiency, this method of
solving the model Hamiltonian enables us to label
the states by the irreducible representation which
they belong to. In the succeeding sections of this
paper, we present the technique of solving spin
Hamiltonians using spin and spacial symmetry adaptation. Then we discuss a theoretical approach to
compute molecular magnetic anisotropy in SMMs.
2.
Solving exchange Hamiltonian
The constant MS basis can be trivially constructed
from the Fock space by choosing those states whose
total MS value corresponds to the desired value. It is
also quite straightforward to set up the Hamiltonian
matrix in this basis and solve for a few low-lying
states in cases where the Hilbert space is spanned by
a few hundred million states.8 It is computationally
difficult to solve systems with magnetic frustration.
In such systems, the ground state spin is often not
predictable and one needs to obtain the lowest state
in each total spin subspace to fix the spin of the
ground state. Besides, it is also numerically difficult
to achieve convergence to nearly degenerate eigenstates with different spin values, unless they can be
dispersed into orthogonal Hilbert spaces. We can
partially alleviate this problem by employing the
parity symmetry of the exchange Hamiltonian. This
symmetry can break the MS = 0 space into even and
odd total spin subspaces.9 Since in most cases, the
lowest excited state usually has a spin which is one
different from that of the ground state, this symmetry makes it easy to obtain the spin gaps accurately.
Construction of spin adapted configuration state
functions (CSFs) has been a long standing problem
of interest. The CSFs are simultaneous eigenstates
of total S2tot and Sztot and setting up the Hamiltonian
matrix in this basis leads to matrices of smaller size
besides allowing automatic labelling of the states by
the total spin. Furthermore, the eigenvalue spectrum
is enriched, since we can obtain several low-lying
states in each total spin sector. This is in contrast to
obtaining several low-lying states in a given total MS
sector as the latter would have states with total spin
Stot ≥ MS. There are many ways of achieving CSFs.10
The methods which have been in extensive study are
(i) the graphical unitary group approach (GUGA),
(ii) symmetry group graphical approach (SGGA)
and (iii) valence–bond (VB) approach. Complete
factorization of the Hilbert space of the Hamiltonian
using both spin and spatial symmetries has been the
focus of many studies.9 However, exploitation of
spatial and spin symmetries has been possible only
for Abelian point groups. Here we present a general
technique which is a hybrid method based on
Valence–bond basis and the basis of z-component of
the total spin, which is applicable to all types of
point groups and is easy to implement on computer.
But, before that we will briefly discuss a previous
attempt to symmetrize VB basis11 and associated
difficulties.
2.1
Symmetrized VB approach
Its is non-trivial to get simultaneous eigenstates of
total spin and its z-component, since eigenstates of
the Szt ot operator expressed as a product of the eigenstates of all the Szi operators are not simultaneously
eigenstates of the S2tot operator. The situation is further complicated by the fact that in a molecular
magnet, often the spins of all the constituent mag-
Computing magnetic anisotropy constants of single molecule magnets
netic centers, si are not the same. In such a case, the
easiest way of constructing the spin adapted functions is the diagrammatic valence bond (VB) method
based on modified Rumer–Pauling rules.12–15 In this
method, a magnetic site with a given spin ‘si’ is replaced by 2si spin-half objects. To obtain a state
with total spin S from N such spin-1/2 objects from
all the magnetic centers, (N – 2S) of these spin-1/2
objects are singlet spin paired explicitly, subject to
the following restrictions: (1) there should be no
singlet pairing of any two spin-half objects belonging to the same magnetic center (this ensures that
the 2si objects are in a totally symmetric combination16), (2) a total of 2S spin-half objects are left
unpaired, (3) when all the spin-half objects are arranged at the vertices of a regular polygon with
number of vertices equal to number of spin-half
objects, N, and straight lines are drawn between spin
paired vertices, there should be no intersecting lines
in the resulting diagram and (4) when all the spinhalf objects are arranged on a straight line and lines
are drawn between spin paired objects, these lines
should not enclose any unpaired spin-1/2 object.
These rules follow from the generalization of the
Rumer–Pauling rules to objects with spin greater
than 1/2 and total spin greater than zero. The set of
diagrams which obey these rules would hence forth
be called ‘legal’ VB diagrams. Some legal VB diagrams are shown in figure 2.
A line in the VB diagram between two spin –1/2
objects i and j corresponds to the state (αiβj–βiαj)/√2
where we choose α to correspond to |↑⟩ and β to |↓⟩
orientations of the spin. The phase convention
assumed is that the ordinal number ‘i’ is less than
the ordinal number ‘j’. The 2S spin-1/2 objects k1 k2
Figure 2. The top VB diagram shows spin pairings to
yield a total spin Stot = 0 state from ten spin-1/2 objects,
constituent elementary spins of two spin 1 and two spin
3/2. Its bit representation corresponds to unique integer
I = 856. The bottom VB diagram shows a Stot = 1 state,
the corresponding unique integer is I = 888.
825
k3 ⋅⋅⋅ k2S which are left unpaired can be taken to
represent the state with MS = S given by
α k1α k2 α k3 ... α k2S . VB states corresponding to other
MS value for this state with spin S, can be obtained
by operating the Stot operator on the state by the required number of times. The VB state corresponding
to a given diagram is a product of the states representing the constituent parts of the diagram. On a
computer, a ‘legal’ VB diagram of any spin can be
uniquely represented by an integer of N bits with a
bit state ‘1’ at a site representing the beginning of a
singlet line and a state ‘0’ the ending of a singlet
line. The unpaired spins are also represented as onebits. Figure 2 also shows bit representation of typical VB diagrams.
To spatially symmetrize a VB basis, it is necessary to know the result of a symmetry operator operating on a legal VB diagram. Unfortunately this leads
to ‘illegal’ VB diagrams, decomposing them into
‘legal’ ones is computationally demanding. An
example of this is shown in figure 3.11 In practice,
the VB space is broken down into smaller invariant
subspaces by the actions of group operations and
then disjoint invariant subspaces are identified.
Within each disjoint invariant space, a symmetrized
linear combination of the VB basis is constructed by
the application of projection operator. However, the
structure of the invariant spaces is very complex and
constructing disjoint invariant spaces is not simple.11
While the number of linearly independent symmetry
combinations of a given representation is known a
priori, the actual linear combinations are obtained by
carrying out Gram–Schmidt orthonormalization of
the projected states. However, since the VB diagrams
are not orthogonal the orthonormalization process is
both computationally involved and time consuming.
Furthermore, in case of molecular magnets containing magnetic ions with spin greater than half, the
exchange operator between such high spin centers
also generates ‘illegal’ VB diagrams as it involves
non-nearest neighbour exchange interactions between
constituent elementary spins.9,14,15
In view of these difficulties, a fully symmetrized
VB approach to solving Heisenberg exchange Hamiltonian particularly in the context of molecular
magnets is not feasible.
2.2 Hybrid method based on VB basis and
constant MS basis
In the constant MS basis, a basis state of an ensemble
of spins s1, s2, … , sN, is represented by a direct
S Ramasesha et al
826
Figure 3. The effect of operation by the C13 symmetry operator about the (1, 8) axis. Top left
shows the initial and final VB diagrams with spin couplings between vertices of the cube
shown as dark lines. Bottom left shows the same states as spin couplings between vertices of a
regular octagon. The resultant is an ‘illegal’ diagram. Decomposing the resultant to ‘legal’ VB
diagrams yields a sum of five VB diagrams shown on the right, with spin couplings between
vertices on a cube.
product of the ms states of each spin such that the
total MS = ∑mi. By construction the states are
orthonormal. A VB diagram with p singlet lines can
be broken up into a linear combination of 2p basis
states in the constant MS basis. In conversion of VB
diagrams to constant MS basis, each singlet line
gives two states; in one, the site at which a singlet
line begins is replaced by an α spin and the site at
which the line ends is replaced by a β spin with
phase factor +1 and in the other state, spins are
reversed with an associated phase of –1. However,
there is a normalization constant wi, which follows
from Clebsch–Gordan coefficients, given by,
(2 si )!
⎡
⎤
wi = ⎢
⎥
⎣ ( si + mi )!( si − mi )! ⎦
−1/ 2
,
(1)
for a site with composite spin si in state mi, given
by, mi = (ni↑ – ni↓)/2. Here, ni↑ is the number of
m = +1/2 and ni↓ is the number of m = −1/2 constituent spins at the ith site.16 Without loss of generality,
we can assume that the MS value of the VB diagram
is also S. It is computationally straightforward to
express a state in the VB basis as a linear combination of states in the constant MS basis. We initialize
the coefficients in the constant MS basis to zero. We
then decompose, sequentially, each VB diagram into
constant MS states and update the coefficient of the
basis state of corresponding MS by adding to it the
VB coefficient times the product of Clebsch–Gordan
factors with appropriate phases. The matrix relating
the VB basis states to constant MS basis states, C, is
a V × M matrix, where V is the dimensionality of the
VB space and M that of the constant MS space.
In the constant MS basis, we can get the matrix
representation of a symmetry operator, R̂ of the
point group in the chosen MS space by operating
with R̂ on each state and searching for the resulting
state in the list of MS basis states. Each basis state in
this representation is carried over to another basis
state by a symmetry operation of the point group.
Thus, the matrix RM though an M × M matrix contains only one non-zero element in each row; this
makes manipulations with this matrix computationally fast. The knowledge of the C and the RM matrices give the effect of operating by the symmetry
operator R̂ on a VB state as a linear combination of
the constant MS basis states via the matrix BR =
CRM.
The projection operator for projecting out the
basis states on to a chosen irreducible representation
of the point group Γ is given by,
PΓ = ∑ χ Γirr ( Rˆ ) Rˆ ,
(2)
Rˆ
where χ Γirr ( Rˆ ) is the character under the symmetry
operation R̂ in the character table of the point group
of the system.17 In our approach, one can easily get
the effect of projection operator on VB states,
expressed in constant MS basis. The matrix representation of PΓ, obtained in the mixed VB and constant
MS basis is given by,
QΓ = ∑ χ Γirr ( R) BR ,
(3)
R
where, QΓ is a V × M matrix. However, the rows of
the matrix QΓ are not all linearly independent. The
Computing magnetic anisotropy constants of single molecule magnets
exact dimension, VΓ, of the Hilbert space spanned by
the system in the irreducible representation Γ can be
known a priori, which is given by,
VΓ = ( dΓ /h) ∑ χ ( Rˆ ) χ Γirr ( Rˆ ),
(4)
Rˆ
where dΓ is the dimensionality of the irreducible representation Γ, h is the number of elements in the
point group and χ ( Rˆ ) is the reducible character for
the operation R̂ . The VΓ × M projection matrix, PΓ
of rank VΓ is obtained by Gram–Schmidt orthonormalization of the rows of the matrix QΓ until VΓ
orthonormal rows are obtained. The M × M Hamiltonian matrix, HM is constructed in the constant MS
basis which is described elsewhere.8 The projected
VΓ × VΓ Hamiltonian matrix in the fully symmetrized
basis is given by,
H Γ = PΓ H M ( PΓ )† ,
(5)
and one could use any of the well-known full diagonalization routines to obtain the full eigenspectrum
or use the Davidson algorithm to get a few low-lying
states of the symmetrized block Hamiltonian in the
chosen spin and symmetry subspace.
We can further reduce dimensions of the blocks of
Hamiltonian corresponding to E, T or higher order
representations, which give degenerate eigenvalues.
In such cases, it is advantageous to work with bases
that transform according to one of the components
of the irreducible representation. This will lead to
unique eigenvalues. To achieve this, we choose an
axis of quantization and project out bases states of
the irreducible representation which are diagonal
about a rotation about the chosen axes. For example,
in the case of the irreducible representation that
transforms as T, we can choose one of the C3 axes as
a quantization axis and project the basis states of the
irreducible representation using (I + C13 + C–31) as the
projection operator. This operator projects states that
transform as the Y10 component of the three fold degenerate irreducible tensor operator. Similarly, for
the E representation, we could use any of the C2 axis
as a quantization axis and use the projection operator (I + C21) to project out basis states that transform
as one of its components.
2.2a Discussion on hybrid method: Computationally, this hybrid method involves few more steps
than that of only constant MS method.9 Firstly, we
827
need to construct the C matrix, whose ith row contains the coefficients of the constant MS functions
appearing in the ith VB basis function. This is a
pretty fast step as the constant MS states are an
ordered sequence of integers and a VB state with n
lines is a linear combination of 2n constant MS functions. Secondly, in the hybrid approach, the computation of the Q matrix involves the matrix
multiplication,
⎛
C ⎜ ∑ χ Γirr ( R ) RM
⎝ R
⎞
⎟ = CRM′ .
⎠
The number of arithmetic operations involved is
however very small, since both C and RM′ are sparse
matrices. Gram–Schmidt orthonormalization for obtaining the projection matrix PΓ from the Q matrix
in the hybrid approach and from R M′ matrix in constant MS approach – both involve similar computational effort. The advantage of fewer orthonormal
rows being sought for PΓ in the hybrid approach
compared to the projection matrix in the constant MS
approach is compensated by the loss of sparseness
of Q matrix in the hybrid approach. Computation
of the eigenvalues in the constant MS approach is
slower than in the hybrid approach, since
( D(Γ M S ) > D(Γ S )) for most S, where D(ΓS) is the
dimensionality of the space of the irreducible representation Γ with spin S and D (Γ M S ) is similarly the
dimension of the space Γ with constant MS. The
memory required for the hybrid approach is not very
different from that of constant MS approach since the
matrices though smaller in the hybrid approach, are
slightly denser. The only additional array required
to be stored in the hybrid approach is the sparse C
matrix.
The major advantage of the hybrid approach is
that we can obtain a far richer spectrum, since we
are targeting each spin sector separately, unlike in
the constant MS approach. Thus, if we can obtain,
say 10 states in each S sector of the 2n spin-1/2
problem, we would have 10(n + 1) unique states
compared to the constant MS technique where many
of these spin states would be repeated in different
MS sectors. This approach is applicable not only to
spin systems, it can be easily extended to electronic
systems as well. Indeed, in table 1, we show the
break-up of the many body states of a π-system,
made up of sp2 carbon atoms placed at the vertices
of an icosahedron, into various subspaces of Irmh for
different total spin values.
S Ramasesha et al
828
Table 1. Dimensionalities of various subspaces of half-filled icosahedral
electronic system.
Stot →
Γ↓
0
1
2
3
4
5
6
Ag
T1g
T2g
Gg
Hg
Au
T1u
T2u
G2u
Hu
2040
16602
16602
30272
47940
1852
17080
17080
30160
46880
3128
28821
28821
50932
79305
3188
28686
28686
50992
79680
1684
14625
14625
26236
41255
1644
14700
14700
26176
40980
382
3261
3261
5880
9220
348
3372
3372
5888
38
309
309
568
900
40
294
294
560
3
6
6
16
40
0
18
18
16
1
0
0
0
0
0
0
0
0
Tot Dim→226512 382239
196625
44044
4212
143
1
method by applying it to the exchange Hamiltonian
of the molecular magnet Cu6Fe8 which has cubic
symmetry.
Figure 4. Schematic of Cu6Fe8 cluster. Filled and open
circles correspond to Fe and Cu (both spin-1/2) sites respectively. Lines represent the exchange coupling between
various spin sites.
In this approach, we have demonstrated, how by
combining the ease of spin symmetry adaptation of
the VB method with the spatial symmetry exploitation of the constant MS methods, we can devise a
scheme which is fully spin and spatial symmetry
adapted. This has been possible because of the ease
of transformation of the VB basis to the constant MS
basis. The method described here can easily be extended to fermionic systems and should provide a
significant improvement for obtaining exact eigenstates of spin conserving model Hamiltonians. In the
next section, we have demonstrated the power of the
2.2b Application to Cu6Fe8: We have applied the
above method to model the susceptibility behaviour
of the molecule [(Tp)8(H2O)6Cu6IIFe8III(CN)24]
(ClO4)4⋅12H2O⋅2Et2O,18 where Tp stands for hydrotris (pyrazolyl) borate (figure 4). In this molecule,
both CuII and FeIII ions are in spin-1/2 state. The
eight FeIII ions are at cube corners and the six CuII
ions are on the outward perpendicular to the face
centers of the cube. Each CuII ion is connected to the
four nearest FeIII ions via ferromagnetic exchange
interactions. There are no Fe–Fe or Cu–Cu interactions. This system has a very high symmetry of the
cube and incorporates all the complexities that can
be encountered in the application of our technique.
From the susceptibility data, the strength of the
exchange interaction J, was estimated to be 30 cm–1.18
The dimensions of the various subspaces are given
in table 2. Using the hybrid method, we have broken
down the space in each total spin sector into basis
states that transform as different irreducible representations of the cubic point group.
The dimensionalities of the various subspaces are
shown in table 3. The subspaces transforming as the
E or T representations are further broken down as
we discussed before. We have setup the Hamiltonian
matrix in each of the subspaces and obtained all the
eigenvalues. We have also used a constant MS basis
and using the full cubic symmetry, factored the
space into various irreducible representations and
obtained all the eigenvalues in each subspace. From
the eigenvectors, we have computed the total spin of
Computing magnetic anisotropy constants of single molecule magnets
829
Table 2. Dimensionalities of the total spin spaces of a system of 14 spin-1/2
objects. D(S) is the dimensionality of the constant S basis and D(MS) is the
dimensionality of the constant MS basis.
S/MS
D(S)
D(MS)
0
1
2
3
4
5
6
7
429
3432
1001
3003
1001
2002
637
1001
273
364
77
91
13
14
1
1
Table 3. Dimensionalities of various subspaces of the Cu6IIFe8III cluster for irreducible representation.
Stot →
Γ↓
A1g
A2g
Eg
T1g
T2g
A1u
A2u
Eu
T1u
T2u
0
1
2
3
4
5
6
7
6
13
34
78
66
5
17
36
105
69
32
15
90
165
216
19
20
78
180
186
24
19
90
171
186
13
27
84
219
168
24
8
60
99
138
11
15
48
123
111
9
5
26
39
54
2
10
20
66
42
5
0
10
6
21
0
2
6
15
12
1
0
2
0
3
0
1
0
6
0
1
0
0
0
0
0
0
0
0
0
the state. We find a one to one correspondence to
numerical accuracy, between the two sets of calculations. We have also fitted the χT vs T experimental
plot by using the full spectrum of the Heisenberg
Hamiltonian and computing19
3⎡
g 2 F ( J ,T )
⎤
χT = ⎢
,
8 ⎣1 − zJ ′F ( J , T )/k BT ⎥⎦
(6)
where we have taken the g factor to be 2⋅1, the ferromagnetic exchange constant J to be 27⋅2 cm–1.
Here, χT is in units of NμB. The function F(J, T) is
given by,
F ( J ,T ) =
∑ M S2 exp[− E0 (S , M S )/kBT ]
S ,M S
∑ exp[− E0 (S , M S )/kBT ]
,
(7)
S ,M S
with E0(S, MS) being the eigenvalue of the sum of
exchange Hamiltonian and the magnetic anisotropy
term DSZ2 and zJ′ is the intermolecular exchange
interaction. Here we have assumed that the molecular anisotropy is along the global z-axis, and this
term is treated as a perturbation to the exchange
Hamiltonian in (1). In figure 5, we show the fit of
the experimental data to the model.
Figure 5. Fit of the χT vs T plot for the Cu6IIFe8III
cluster. The best fit parameters are, J = 27⋅2 cm–1 (ferromagnetic), zJ′ = –0⋅008 cm–1 (antiferromagnetic), D =
–0⋅15 cm–1 and g = 2⋅1.
3.
Computing magnetic anisotropy in SMMS
The magnetic anisotropy is given by the general
Hamiltonian,
Hˆ D = SˆM .D ( M ) .SˆM ,
(8)
830
S Ramasesha et al
where SˆM corresponds to the spin operator for the
total spin of the molecule and D(M) is the magnetic
anisotropy tensor of the molecule. In practice, this
D(M) tensor is diagonalized and the eigenvalues of
the tensor give magnetic anisotropy in three mutually perpendicular principal directions while the
eigenvectors corresponds to the direction of orientation of the principal axes with respect to the molecular frame. If, DMXX, DMYY and DMZZ are the molecular
anisotropies along the three principal directions such
that DMXX + DMYY + DMZZ = 0, we can define two parameters, DM and EM such that,
1 M
M
M
DM = DZZ
− ( DXX
+ DYY
)
2
1 M
M
EM = ( DXX
− DYY
),
2
(9)
where DM and EM are called the axial and rhombic
anisotropies respectively. Using these two parameters, the Hamiltonian in (8) can be rewritten as,
H M = DM ( SˆZ2 − S ( S + 1)) + EM ( Sˆ X2 − SˆY2 ).
(10)
The magnetic anisotropy could be of relativistic or
dipolar origin. In systems like conjugated polymers,
the anisotropy arises due to the dipolar interactions,
whereas in SMMs containing transition metal ions,
relativistic effects dominate and the dipolar contribution to anisotropy is two to three orders of magnitude smaller. There have been several theoretical
methods to compute the molecular magnetic anisotropy in molecular magnets. The first method involves computing the DM and EM values by tensoral
summation of the anisotropies of the constituent
transition metal centers.20–22 However, this leads to
molecular anisotropy values which are independent
of the total spin state of the molecule. The next
method involves computation of D(M) tensor using
the effective mean-field potential ϕ (r) obtained
from density functional theory (DFT) with desired
S Zt otal of the cluster. Treating spin orbit (SO) operaˆ ( r )] as a perturbation, the D(M) tensor
tor, Sˆ.[ pˆ × ∇Φ
is obtained.23–27. However, DFT methods do not conserve either the total spin or the site spins. Thus, it is
not possible to obtain a pure spin eigenstate. Moreover, mean-field theories do not give correct values
of spin–spin correlations. There have been methods
to obtain the single-ion anisotropy values using restricted configuration interaction approach.28 Bencini
and Gatteschi developed a perturbative technique to
obtain the anisotropy values of bi-nuclear systems.
This method is analytical and cannot be employed
for larger systems like Mn12Ac and Fe8 clusters.
Here, we present a general method to compute
magnetic anisotropy of large cluster of ions with
arbitrary spins in any given total spin state. We use a
spin-exchange Hamiltonian to describe the cluster,
unlike the all electron Hamiltonian that is employed
in DFT studies. We obtain desired exact eigenstates
of the exchange Hamiltonian by methods described
above and using the magnetic anisotropic interactions as a perturbation to compute the molecular anisotropy constants in the desired eigenstates, in first
order in perturbation. The input parameters required
in our study are the local single ion anisotropies and
the exchange constants of the exchange Hamiltonian. Our study can yield the anisotropy values for
different total spin states as well as for different
states with the same total spin. In the next section
we describe the method in detail. In the following
section we present the results of our studies on the
two SMMs, Mn12Ac and Fe8 and finally we summarize our studies.
3.1
Methodology
We treat the exchange Hamiltonian between magnetic centers in the SMMs as the unperturbed Hamiltonian,
Hˆ 0 = ∑ J ij Sˆi .Sˆ j ,
(11)
⟨ ij ⟩
where ⟨ji⟩ runs over all pairs of centers in the model
for which the exchange constant is non-zero, Sˆi is
the spin on the ith magnetic center. In SMMs such as
Mn12Ac the spins at all the magnetic centers are not
the same and the exchange interactions are shown in
figure 6. H0 can be solved exactly for a few lowlying states in a chosen spin sector by using methods
that have been described above.9
The general anisotropic Hamiltonian for a collection of magnetic centers Ĥ1′ is given by,
1
Hˆ 1′ = ∑ Dij ,αβ Sˆiα Sˆ βj ,
2 i , j ,α , β
(12)
where the indices i and j run over all the magnetic
centers and α and β run over x, y and z directions of
the ion. The contributions to inter-center anisotropy
Computing magnetic anisotropy constants of single molecule magnets
constant arise due to dipolar interaction between the
spins on the two centers as well as due to relativistic
effects. In the former, Dij,αβ is given by,
Dij ,αβ
ℜij2 δ αβ − 3Rij ,α Rij , β
1
,
= g 2 μ B2
2
Rij5
(13)
where ℜij ( R ij ) is the vector (distance) between the
magnetic centers i and j, g is the gyromagnetic ratio
and μB is the electronic Bohr magneton; the expectation value in (13) is obtained by integration over
spatial coordinates.29 Approximating the expectation
values of the distances by the equilibrium distances,
the Dij,αβ in (13) and by computing the necessary
spin–spin correlation functions, we can obtain the
molecular Dαβ(M) tensor.30 The eigenvalues of this
matrix give the principal anisotropy values and
imposing the condition of zero trace of the matrix
yields molecular magnetic anisotropy constants due
to spin–spin interactions. The magnetic anisotropies
computed assuming only spin dipolar interactions
yielded negligibly small values in comparison to the
experimentally observed DM = –0⋅7 K and –0⋅28 K
for Mn12Ac and Fe 8 SMMs respectively, in the
S = 10 ground state.31,32 Thus, we need to compute
the magnetic anisotropy of SMMs from the singleion anisotropies of the constituent magnetic centers
since, in SMMs relativistic effects dominate due to
the presence of transition metal ions. The relativistic
Figure 6. Schematic of possible exchange interactions
in Mn12Ac SMM. The peripheral Mn(III) ions represented
by blue circles correspond to spin-2 sites and those represented by yellow circles are the core Mn(IV) ions each of
spin-3/2. Js are the strength of superexchange interaction
with J1 = 215 K, J2 = J3 = 85⋅6 K, J4 = –64⋅5 K.6
831
interactions are short-ranged, they fall off as r–3.
Hence in (12), we can ignore inter-site terms and
replace Ĥ1′ by Ĥ1 given by,
Hˆ 1 =
∑ Di,α Sˆiα Sˆiβ .
(14)
i ,α , β
Since, the local anisotropies of the individual ions
can have their own set of principal axes, we need to
project out the components of anisotropy on to the
laboratory axes. This modifies the (14) to
Hˆ 1 =
∑
i ,α , β , l , m
Ci ,lα Ci , mβ Di ,αβ Sˆiα Sˆiβ ,
(15)
where, Ci,lα is the direction cosines of αth coordinate of the local axis of the ith magnetic center with
the lth coordinate of the laboratory frame. Since, the
Hamiltonians in (8) and (15) are equivalent, we can
equate the matrix elements ⟨n, S M , S | Hˆ 1 |n, S M , S ′⟩
and ⟨ S M , S |Hˆ D |S M , S ′⟩ and for any pair of eigenstates of the exchange Hamiltonian in (11); M and
M′ are z-component of total spin in a state n with
spin SM in which we are interested. Calculating these
matrix elements for Hˆ D is straightforward from the
algebra of spin operators. However, evaluation of
the matrix elements of Ĥ1 between eigenstates of
Ĥ 0 , requires a knowledge of the matrix elements of
different products of site spin operators (such as
Sˆix Sˆ jy ). For this we need eigenstate of Ĥ 0 ,
| n, S , M ⟩, for different M values and these are obtained by using the ladder operators corresponding
to spin S.
Given a S value, the above condition would give
rise to (2S + 1)2 equations, while the tensor D(M), has
only nine unknowns corresponding to the nine components of the second rank tensor. Thus, for the
Mn12Ac system, with ground state spin of 10, there
would be 441 equations and we have more equations
than unknowns. However, we could take any nine
equations and solve for the components of the tensor
D(M) and we would get unique values of the components. This is guaranteed by the Wigner–Eckart
theorem and we have also verified this by solving
for the D(M) tensor from several arbitrarily different
selections of the nine equations.
3.2
Results and discussion
We have computed DM and EM values for both
Mn12Ac and Fe8 systems by systematically rotating
the local anisotropies of the spin centers. We have
832
S Ramasesha et al
used the single-ion anisotropy values quoted in the
literature for the magnetic ions in similar ligand environments.33–36 The local ion axis referred to as x, y
and z and the laboratory axis denoted by X, Y and Z
are shown in figure 7. The laboratory frame can be
arbitrarily chosen since, while computing the molecular anisotropy, we diagonalize the anisotropy tensor and obtain the anisotropy values along the three
principal directions. The principal axis of the molecule is unique and it does not depend on the orientation of the laboratory axis.
In order to obtain the molecular anisotropy as a
function of the angle θ which the z-axis of the ion
makes with the laboratory Z-axis, we rotate the singleion orientation with respect to the laboratory frame.
The orientation of z-component of the single-ion
anisotropy at every site is shown in figures 8, 9
and 10 (schemes 1, 2 and 3) for Mn12Ac and in figures 14, 15 and 16 (schemes 4, 5 and 6) for Fe8 sys
tems respectively. First the z-axis ( z ) of the ion is
fixed and then x is obtained by Gram-Schmidt orthogonalization procedure. Though the choice of this
vector is arbitrary in a plane perpendicular to z-axis,
we have fixed the direction of x such as to have
maximum projection along a M–O (M = Mn, Fe)
bond in Mn12Ac as well as in Fe8 (figures 7 and 16).
If O is the vector connecting a M site and
a neighbouring O ion, then we obtain x using the
relation,
x = O − (O.z ) z .
(16)
Finally, the y-axis of the ion is obtained using the
relation, y = z × x. These three mutually orthogonal
vectors are then normalized to obtain the orthonormal set of coordinate axes x, y and z of the ion center. These single-ion axes of a given site i can be
projected on to the laboratory frame through the
direction cosines, Ci,αβ, where, α = X, Y, Z and β = x,
y, z (17).
x = Ci,XxX + Ci,YxY + Ci,ZxZ
y = Ci,XyX + Ci,YyY + Ci,ZyZ
z = Ci,XzX + Ci,YzY + Ci,ZzZ.
(17)
3.2a Magnetic anisotropy in Mn12Ac SMM: We
have first obtained the ground state and few excited
states of the Mn12 Ac system by exactly solving the
Figure 8. Schematic diagram showing the directions of
local anisotropy in Mn12Ac. The single-ion anisotropies
of all the Mn ions are directed along the laboratory Z axis
(scheme 1).
Figure 7. Schematic of local (x, y, z) and laboratory
(X, Y, Z) coordinate axes in Mn12Ac. The blue, green and
red spheres correspond to Mn(III) (spin-2), Mn(IV) (spin3/2) and oxygen ions respectively. The arrows indicate
the Mn–O bonds on which the chosen local x-axis has
maximum projection.
Figure 9. Schematic diagram showing the directions of
local anisotropy in Mn12Ac. The z-component of the
single-ion anisotropies of all the Mn(III) ions are inclined
at an angle θ to the laboratory Z and while that of the
Mn(IV) ions are kept fixed at ~48° (scheme 2).
Computing magnetic anisotropy constants of single molecule magnets
unperturbed Hamiltonian given in (11), using the
exchange interactions shown in figure 6.6 The
ground state of the system corresponds to total spin
10 with a total spin 9 first excited state at 35⋅1 K
from the ground state and a S = 8 second excited
state at 60⋅4 K from the ground state. The molecular
magnetic anisotropy is computed using the singleion anisotropy values of –5⋅35 K and 1⋅226 K
respectively for Mn(III) and Mn(IV) ions, obtained
from the literature. We have also introduced transverse anisotropy of 0⋅022 K and 0⋅043 K for Mn(III)
and Mn(IV) ions respectively. We have studied the
variation of molecular anisotropy as a function of
orientation of the local anisotropies by rotating the
local D tensor around the molecular Z-axis (refer
figures 8–10). In scheme 1, all the single–ion z axes
are pointed parallel to the laboratory Z direction
while in scheme 2 the orientation of the local anisotropies of the core Mn(IV) ions are along the line
joining the ion and the molecular center (~48° from
the laboratory Z-axis), while that of the Mn(III) ions
is kept fixed at an angle θ from the Z axis. The orientation of the local anisotropy of Mn(III) ions for
which we get the best agreement with experiments
corresponds to θ ~ 17°. In scheme 3, we have fixed
the orientation of single-ion anisotropy along the
laboratory X – Y plane. We have studied the variation in the molecular anisotropies in these schemes
for ground and excited eigenstates and presented the
Figure 10. Schematic diagram showing the directions
of local anisotropy in Mn12Ac. The z-component of the
single-ion anisotropies of all the Mn ions are directed
along the plane perpendicular to the laboratory Z axis
(scheme 3).
833
results in table 4. We note that when the local anisotropies are systematically varied, there is a very
large variation in the molecular anisotropy as a function of the local orientation (figure 11). The variation of DM with θ follows the equation DM(S) = DM0
(S)(3cos2θ − 1). We find this in all the eigenstates of
Mn12Ac that we have studied. The molecular anisotropy values are different in different spin eigenstates. This may be rationalized from the fact that
the energy gaps between the ground and the excited
states are large as a consequence of which the spin
correlations in these states are very different. From
the eigenvectors of the D(M) matrix, we find that the
choice of our laboratory frame we have chosen
is very close to the principal axis of the molecular
system in all cases.
We have also studied the role of magnetic orientations of the core Mn(IV) ions (s = 3/2) and the
crown Mn(III) ions (s = 2) in determining the molecular anisotropies by fixing the single ion anisotropy
directions of the crown Mn(III) ions fixed at 0° and
rotating only the orientation of the anisotropy direc-
Figure 11. Variation of DM as a function of θ, the angle
the z-component of local anisotropy of Mn(III) ions
makes with the laboratory Z-axis in eigenstates with total
spin 10, 9 and 8. The orientation of Mn(IV) ions is kept
fixed at ~48° from the molecular Z-axis. The curve with
filled circles correspond to the variation of DM, when the
local anisotropies of the core Mn(IV) ions only are
rotated and those of Mn(III) ions are fixed along the
Z-axis. The variation of DM with θ follows the equation
DM(S) = DM0 (S)(3cos2θ − 1), with all DM0 (10) = –0⋅40,
DM0 (9) = –0⋅34, DM0 (8) = –0⋅25. Schemes 1, 2 and 3 correspond to θ = 0°, 17° and 90°. Best fit for the experimental
DM value in the Stotal = 10 state corresponds to θ ~ 17°.
834
S Ramasesha et al
Table 4. DM values of ground and excited states of Mn12Ac under various schemes
in K. For scheme 2, we have presented the DM values only for θ ~ 17° for which the
DM value of the ground state matches with the experimentally observed value.
DM (K)
State
Scheme 1
Scheme 2
Scheme 3
Ground state (S = 10)
First excited state (S = 9) Eg = 35⋅1 K
Second excited state (S = 8) Eg = 60⋅4 K
–0⋅8138
–0⋅6722
–0⋅5009
–0⋅7083
–0⋅6105
–0⋅4264
0⋅4075
0⋅3449
0⋅2464
Table 5. Energy gaps (Δ) and D0 values for the S = 10 ground state corresponding to different sets of parameter values in Mn12Ac.
S. No.
1
2
3
4
5
J1 (K)
J2 (K)
J3 (K)
J4 (K)
Δ (K)
D0 (K)
215
215
215
215
215
85
85
85
85
85
85
85
64⋅5
45
–85
–64⋅5
–85
–64⋅5
–45
–45
35⋅1
67⋅0
72⋅7
80⋅0
224
–0⋅40
–0⋅43
–0⋅46
–0⋅49
–0⋅58
Mn12Ac, the SX2 − SY2 term of the anisotropy Hamiltonian does not commute with the D2d molecular point
group symmetry and hence EM is zero. To investigate the variation of DM0 with the exchange parameters of the unperturbed Hamiltonian, we have
computed the magnetic anisotropy of Mn12Ac in the
ground state using five different sets of exchange
constants.6 In each case, the ground state has spin
Stotal = 10 and the first excited state corresponds to
S = 9; but the gap to the lowest excited spin state
varies (table 5). We see that there is a steady
increase in |DM0 | with increasing gap (see figure 12).
Figure 12. Variation of |DM0 | as a function of energy gap
(Δ in K) for the Stotal = 10 ground state of Mn12Ac for
parameters listed in table 5, DM(θ) is given by –|DM0 |
(3cos2θ − 1).
tions of the core Mn(IV) ions systematically. The
variation of DM for the S = 10 ground state as a function of rotation of the local anisotropies of the core
Mn(IV) ions is shown in figure 11. We observe that
the molecular anisotropy is insensitive to the orientations of the core Mn(IV) ions while the orientation
of the crown Mn(III) ions control the variation of
the molecular magnetic anisotropy in Mn12 Ac. In
3.2b Magnetic anisotropy in Fe8 SMM: We have
also computed the values of molecular anisotropy
for the Fe8 SMM. First we solve the unperturbed
Hamiltonian in (11) by using exchange parameters,
J1 = 150 K, J2 = 25 K, J3 = 30 K, J4 = 50 K (figure
13).6 The ground state of the system corresponds to
total spin S = 10 with a S = 9 state at 13⋅56 K, a
S = 9 state at 27⋅28 K and a S = 8 state at 28⋅33 K
above the ground state. We have computed the magnetic anisotropy of Fe8 in three different schemes
(schemes 4, 5 and 6, figures 14, 15 and 16) similar
to Mn12Ac, taking the single ion axial and rhombic
anisotropy values for Fe(III) centers to be 1⋅96 K
and 0⋅008 K respectively. We have studied the
variation in the molecular anisotropies as a function
of orientation of local anisotropy in these schemes
for ground and the excited eigenstates (figure 17).
Computing magnetic anisotropy constants of single molecule magnets
We have presented the DM values for the ground and
the excited spin states under various schemes in
table 6. We note that the molecular anisotropy
changes significantly when the local anisotropies are
systematically varied (figure 17), in all the eigenstates that we have studied. We also note that given
the orientations of local anisotropies, the actual molecular anisotropy values are not very different in
different spin eigenstates, since the energy gaps
between the ground and the excited states are small
which in turn imply that the spin correlations in
these states are not significantly different. The ori-
835
entation of the local anisotropy centers for which we
get the best agreement with experiments (DM =
–0⋅28 K) corresponds to θ ~ 82°.35,36 As with Mn12 Ac,
we find that the laboratory frame we have chosen is
very close to molecular axis in all the cases. In case
of Fe8 cluster, the D2 symmetry commutes with the
Hamiltonian in (10) and allows for a non-zero EM
term. The variation of EM as a function of θ is shown
in figure 18, the value of EM for which DM has the
best fit is 0⋅017 K compared to the experimental estimate of 0⋅046 K obtained from High-frequency
EPR measurements.32 In this case also, we have
explored the variation of DM0 with the exchange constants in the unperturbed Hamiltonian (11).6 Unlike
Figure 15. Schematic diagram showing the directions
of local anisotropy in Fe8. The z-component of the singleion anisotropies of all the Fe(III) ions are inclined at an
angle θ to the laboratory Z-axis (scheme 5).
Figure 13. Schematic of exchange interactions in Fe8
SMM. Js are the strength of superexchange interaction
with J1 = 150 K, J2 = 25 K, J3 = 30 K, J4 = 50 K.9
Figure 14. Schematic diagram showing the directions
of local anisotropy in Fe8. The single-ion anisotropies
of all the Fe(III) ions are directed along the laboratory
Z-axis (scheme 4).
Figure 16. Schematic diagram showing the directions
of local anisotropy in Fe8. The z-component of the singleion anisotropies of all the Fe(III) ions are directed along
the plane perpendicular to the laboratory Z-axis (scheme
6). The arrows indicate the Fe–O bonds on which the
chosen local x-axis has maximum projection.
836
S Ramasesha et al
Table 6. DM values of ground and excited states of Fe8 under schemes 4, 5 and 6 in K. For
scheme 5, we have presented the DM values only for θ = 82⋅2° for which the DM value of the
ground state matches with the experimentally observed value. Experimental DM values are
given in parenthesis and the corresponding references in square brackets.
DM (K)
State
Scheme 4
Scheme 5
Scheme 6
0⋅6030
0⋅5821
0⋅5877
0⋅5503
–0⋅2867 (0⋅28)32
–0⋅2763 (–0⋅27)37
–0⋅2790 (–0⋅27)37
–0⋅2607
–0⋅3033
–0⋅2923
–0⋅2952
–0⋅2758
Ground state (S = 10)
First excited state (S = 9) Eg = 13⋅56 K
Second excited state (S = 9) Eg = 27⋅28 K
Third excited state (S = 8) Eg = 28⋅33 K
Table 7. Energy gaps (Δ) and D0 values for the S = 10 ground state corresponding to different sets of parameter values in Fe8.
S. No.
1
2
3
4
J1 (K)
J2 (K)
J3 (K)
J4 (K)
180
150
195
201
153
25
30
36⋅2
22⋅5
30
52⋅5
58⋅3
52⋅5
50
22⋅5
26⋅1
Figure 17. Variation of DM in Fe8 cluster as a function
of θ, the angle the z-component of local anisotropy of
Fe(III) ions makes with the laboratory Z-axis. All the
plots can be fitted to DM0 (S)(3cos2θ − 1), with DM0 (10) =
0⋅3, DM0 (9) = 0⋅29, DM0 (8) = 0⋅275. DM0 (S) is almost independent of S. Best fit for the experimental DM value in the
Stotal = 10 state corresponds to θ ~ 82°.
with Mn12 Ac, we find that DM0 is almost independent
of the excitation gap of the exchange Hamiltonian
(table 7).
Δ (K)
5⋅87
13⋅56
41⋅40
42⋅5
D0 (K)
0⋅30
0⋅30
0⋅30
0⋅30
Figure 18. Variation of EM in Fe8 cluster as a function
of θ, the angle the z-component of local anisotropy of
Fe(III) ions makes with the laboratory Z-axis for scheme 5.
4.
Conclusions
In this paper we presented a hybrid technique based
on constant MS basis and VB basis which adapts
both spin and spatial symmetry of a general point
group. This technique can be used to obtain the
eigenstates of spin and electronic model Hamiltonians with any molecular point group symmetry. We
Computing magnetic anisotropy constants of single molecule magnets
also presented general method to calculate the molecular magnetic anisotropy parameters, DM and EM
for single molecule magnets in a chosen eigenstate
of the exchange Hamiltonian. The molecular anisotropies are computed from the single-ion anisotropies, using first order perturbation theory for
different spin states of the SMMs. We have computed the molecular magnetic anisotropy parameters
of Mn12Ac and Fe8 SMMs in various eigenstates of
different total spin. We have also studied the variation of molecular anisotropy with the orientation of
the local anisotropy of the metal ions. In Mn12 Ac
and Fe8 clusters, we find that the molecular anisotropy changes drastically with the local anisotropy
direction. In Mn12 Ac, the DM value is different in
ground and excited states we have computed, owing
to large difference in spin–spin correlation values.
The molecular anisotropy of Mn12Ac does not
change significantly with the orientation of the local
anisotropy of the core Mn(IV). DM value in Fe 8 is
not very different in ground and excited states
probably due to small energy gaps which implies
similar spin–spin correlations which is in consistent
with the magnetic anisotropies computed for different sets of exchange constants. We find that in
Mn12Ac, the anisotropy constants increase significantly with the gap, while in Fe8 they are almost independent of the gap. In Mn12Ac, the first order
rhombic anisotropy term is zero due to the D2d
symmetry of the molecule while it is non-zero in
Fe8.
References
1. Thomas L, Lionti F, Ballou R, Gatteschi D, Sessoli R
and Barbara B 1996 Nature 383 145
2. Linert W and Verdaguer M 2003 Molecular magnets:
Recent highlights (New York: Springer-Verlag)
3. Miller J S and Drillon M 2003 Magnetism: Molecules
to materials IV (Wiley-VCH Verlag GMBH)
4. Shoji M, Koizumi K, Hamamoto T, Kitagawa Y,
Yamanaka S, Okumura M and Ya-Maguchi K 2006
Chem. Phys. Lett. 421 483
5. Sessoli R, Gatteschi D, Canneschi A and Novak M A
1993 Nature 365 141
6. Raghu C, Rudra I, Sen D and Ramasesha S 2001
Phys. Rev. B64 064419
7. Rudra I, Ramasesha S and Sen D 2002 Phys. Rev.
B66 014441
8. Pati S, Ramasesha S and Sen D 2002 Magnetism:
Molecules to materials IV (eds) J S Miller and M
Drillon (Wiley-VCH)
9. Sahoo S, Rajamani R, Ramasesha S and Sen D 2008
Phys. Rev. B78 054408
837
10. Pauncz R 1979 Spin eigenfunctions: Construction
and use (New York: Plenum Press)
11. Ramasesha S and Soos Z G 1993 J. Chem. Phys. 98
4015
12. Pauling L 1933 J. Chem. Phys. 1 280
13. Pauling L and Wheland G W 1933 J. Chem. Phys. 1
362
14. Soos Z and Ramasesha S 1990 Valence bond theory
and chemical structure (eds) D J Klein and N Trinajstic (New York: Elsevier)
15. Ramasesha S and Soos Z 2002 Valence bond theory
(ed.) D L Cooper (New York: Elsevier)
16. Arovas D, Auerbach A and Haldane F 1988 Phys.
Rev. Lett. 60 531
17. Bishop D 1973 Group theory and chemistry (Oxford:
Clarendon Press)
18. Wang S, Zuo J-L, Zhou H-C, Choi H, Ke Y, Long
J R and You X-Z 2004 Angew. Chem. Int. Ed. 43
5940
19. Kahn O 1993 Molecular magnetism (New York:
VCH Publishers Inc.)
20. Oudar J L and Zyss J 1982 Phys. Rev. A26 2016
21. Bencini A, Benelli C and Gatteschi D 1984 Coord.
Chem. Rev. 60 131
22. Bencini A and Gatteschi D 1990 EPR of exchange
coupled systems (Berlin: Springer)
23. Pederson M and Khanna S 1999 Phys. Rev. B60
9566
24. Pederson M, Kortus J and Khanna S 2002 J. Appl.
Phys. 91 7149
25. Kortus J, Baruah T, Bernstein N and Pederson M
2002 Phys. Rev. B66 092403
26. Takeda R, Mitsuo S, Yamanaka S and Yamaguchi K
2005 Polyhedron 24 2238
27. Takeda R, Koizumi K, Shoji M, Nitta H, Yamanaka
S, Okumura M and Yamaguchi K 2007 Polyhedron
26 2309
28. Neese F and Solomon E 1998 Inorg. Chem. 37 6568
29. Carrington A and McLachlan A 1967 Introduction to
magnetic ressonance: With applications to chemistry
and chemical physics (New York: Harrer and Row
Publishers)
30. Ramasesha S and Soos Z G 1984 Chem. Phys. 91 35
31. Barra A L, Gatteschi D and Sessoli R 2000 Chem.
Eur. J. 6 1608
32. Barra A-L, Debrunner P, Gatteschi D, Schulz C E
and Sessoli R 1996 Europhys. Lett. 35 133
33. Cornia A, Sessoli R, Sorace L, Gatteschi D, Barra
A L and Daiguebonne C 2002 Phys. Rev. Lett. 89
257201
34. Barra A L, Gatteschi D, Sessoli R, Abbati G L, Cornia A, Fabretti A C and Uytterhoeven M G 1997
Angew. Chem. Int. Ed. Engl. 36 2329
35. Accorsi S, Barra A, Caneschi A, Chastanet G, Cornia
A, Fabretti, A C, Gatteschi D, Mortalo G, Olivieri E,
Parenti F, Rosa P, Sessoli R, Sorace L, Wernsdorfer
W and Zobbi L 2006 J. Am. Chem. Soc. 128 4742
36. Heerdt P T, Stefan M, Goovaerts E, Canneschi A and
Cornia A 2006 J. Mag. Res. 179 29
37. Zipse D, North J, Dalal N, Hill S and Edwards R
2003 Phys. Rev. B68 184408